What is a Confidence Interval?
Statistics is a branch of Mathematics that deals with the collection, classification and representation of data. For the students out in the world wondering “what is a confidence interval?” and “why is it used in statistics?”, this article gives you a brief overview of confidence interval definition, confidence interval formula and how to calculate confidence interval. Confidence interval is a type of interval calculation derived from the data observed. It holds the actual value of the unknown parameter. The confidence interval is linked with the confidence level in which the interval calculates the deterministic parameter. Confidence interval definition is based on Standard Normal Distribution where the value of Z is the z- score.
Confidence Interval Definition:
A confidence level is the representation of the proportion or the frequency of the admissible confidence intervals that consist of the actual value of the unknown parameter. It can be defined the other way round that the confidence intervals can be computed using the given confidence level from a limitless level of individual samples, in such a way that the proportion of the range consists of the true value of the factor that will be identical to the confidence level. In general, confidence level is presumed prior to data examination. In most of the confidence interval examples, the confidence level chosen is 95%. However, the confidence level of 90% and 95% are also used in few confidence interval examples.
Confidence Interval Formula:
The computation of confidence intervals is completely based on mean and standard deviation of the given dataset. The formula to find confidence interval is:
CI = \[\hat{X}\] ± Z x (\[\frac{σ}{\sqrt{n}}\])
In the above equation,
\[\hat{X}\] represents the mean of the data
Z indicates the confidence coefficient
α is the indication of the confidence level
σ is the standard deviation
n is the sample space
The value after the plus or minus sign in the formula is called the margin of error.
The confidence interval table gives the values of Z i.e. the confidence coefficient for the corresponding confidence level. The below table gives the values of confidence coefficients for the corresponding confidence level.
Confidence Interval Table:
How to Calculate Confidence Interval?
A series of steps is to be followed to calculate the confidence interval of a given data sample.
Step 1:
Determine the number of observations in the given sample space denoted as ‘n’. Calculate the mean \[\hat{X}\] and standard deviation σ.
Step 2:
Presume a confidence level of either 95% or 99%. Identify the value of Z for the confidence level chosen. The confidence interval table described in the previous subsection to determine the value of Z.
Step 3:
Substitute the determined values in the confidence interval formula.
CI = \[\hat{X}\] ± Z\[\frac{∝}{2}\] x (\[\frac{σ}{\sqrt{n}}\])
Confidence Interval Examples:
A tree consists of hundreds of apples. 46 apples are randomly chosen. The mean and standard deviation of this instance is found to be 86 and 6.2 respectively. Determine whether the apples are big enough or not.
Solution:
Given data:
Mean \[\hat{X}\] = 86
Standard deviation σ = 6.2
Number of observations n = 46
Let us assume the confidence level as 95%
The confidence coefficient from the table is determined as: Z = 1.960
The formula for confidence interval is:
CI = \[\hat{X}\] ± Z x (\[\frac{σ}{\sqrt{n}}\])
CI = 86 ± 1.960 x (\[\frac{6.2}{\sqrt{46}}\])
CI = 86 ± 1.79
The margin error in this problem is 1.79.
All the hundreds of apples are therefore likely to be in the range of 86 + 1.79 and 86 - 1.79
i.e. in the range of 84.21 and 87.79
Fun Facts about Confidence Interval Formula:
Confidence interval is accurate only for normal distribution of population. However, in case of large samples from other kinds of population distributions, the central limit theorem is used to determine the most accurate interval.
Confidence level of 95% should never be misinterpreted that 95% of the sample population lie within the confidence interval.
Confidence interval is not the estimation of the plausible values of the unknown parameter of the population.
If a confidence level is determined to be 95% for a particular experiment, it is not true that the same confidence level is obtained by repeating the experiment.
FAQs on Confidence Interval
1. What is a confidence interval in statistics?
A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. Instead of providing a single number as an estimate (like the sample mean), it gives a lower and upper bound. For example, instead of saying the average student height is exactly 165 cm, a confidence interval might state we are 95% confident the true average height is between 162 cm and 168 cm.
2. What is the formula for a confidence interval and how is it calculated?
The formula for calculating a confidence interval (CI) for a population mean is: CI = X̄ ± Z * (σ/√n). To calculate it, you follow these steps:
- Step 1: Find the sample mean (X̄), the population standard deviation (σ), and the number of observations (n).
- Step 2: Choose a confidence level (e.g., 95% or 99%) and find the corresponding Z-score from a standard table. For 95% confidence, the Z-score is 1.96.
- Step 3: Calculate the margin of error by multiplying the Z-score by the standard error (σ/√n).
- Step 4: Add and subtract the margin of error from the sample mean to find the upper and lower bounds of the interval.
3. What does it really mean for a confidence interval to be 95%?
A 95% confidence level does not mean there is a 95% probability that the true population mean lies within one specific interval. Instead, it refers to the reliability of the estimation method. It means that if you were to take 100 different random samples from the same population and calculate a confidence interval for each sample, approximately 95 of those intervals would contain the true population mean. It's a measure of confidence in the procedure, not in any single result.
4. What are the main factors that affect the width of a confidence interval?
Three primary factors determine the width of a confidence interval:
- Confidence Level: A higher confidence level (e.g., 99% vs. 95%) results in a wider interval because you need a larger range to be more certain it contains the true parameter.
- Sample Size (n): A larger sample size leads to a narrower interval. With more data, your estimate of the mean becomes more precise, reducing the range of plausible values.
- Variability of the Sample (σ): Greater variability (a larger standard deviation) in the sample data results in a wider interval, as the data points are more spread out, leading to less certainty.
5. How is the significance level, such as 0.05, related to a 95% confidence interval?
The significance level, denoted as alpha (α), is the complement of the confidence level. If the confidence level is 95% (or 0.95), the significance level is α = 1 - 0.95 = 0.05. This alpha value represents the probability that the interval will *not* contain the true population parameter. It is the risk you are willing to take of being wrong, and it is the same value used as a threshold in hypothesis testing.
6. Is there a 'good' value for a confidence interval?
There is no single 'good' value for a confidence interval, as its usefulness is context-dependent. A narrower interval is generally preferred because it suggests a more precise estimate of the population parameter. However, achieving a narrow interval often requires a very large sample size. The choice of confidence level (e.g., 95% or 99%) is a trade-off between precision (width) and confidence. In critical fields like medicine, a higher confidence level like 99% might be considered 'better' despite being wider.
7. When should a student use a t-distribution instead of a Z-distribution for a confidence interval?
You should use a t-distribution instead of a Z-distribution under specific conditions. A Z-distribution is appropriate when the population standard deviation (σ) is known OR the sample size is large (typically n > 30). You must use a t-distribution when the population standard deviation is unknown and you are using the sample standard deviation (s) as an estimate, especially when the sample size is small (n < 30).
8. What is a real-world example of how confidence intervals are used?
Confidence intervals are widely used in many fields. In election polling, when a report says a candidate has 45% of the vote with a 'margin of error' of ±3%, they are describing a confidence interval. This means the pollsters are, for example, 95% confident that the true percentage of support for the candidate in the entire population is between 42% and 48%. It helps communicate the uncertainty inherent in using a sample to estimate a population's opinion.
